g H${bV narjm II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS

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1 Series RLH H$moS> Z. 30/2 Code No. amob Z. Roll No. narjmwu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wi-n ð >na Adí` {bio & Candidates must write the Code on the title page of the answer-book. H $n`m Om±M H$a b {H$ Bg àíz-nì _o _w{ðv n ð> 11 h & àíz-nì _ Xm{hZo hmw H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wi-n ð> na {bi & H $n`m Om±M H$a b {H$ Bg àíz-nì _ >31 àíz h & H $n`m àíz H$m CÎma {bizm ewê$ H$aZo go nhbo, àíz H$m H«$_m H$ Adí` {bi & Bg àíz-nì H$mo n T>Zo Ho$ {be 15 {_ZQ >H$m g_` {X`m J`m h & àíz-nì H$m {dvau nydm _ ~Oo {H$`m OmEJm & ~Oo go ~Oo VH$ N>mÌ Ho$db àíz-nì H$mo n T> Jo Am a Bg Ad{Y Ho$ Xm amz do CÎma-nwpñVH$m na H$moB CÎma Zht {bi Jo & Please check that this question paper contains 11 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 31 questions. g H${bV narjm II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS SET-2 Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at a.m. From a.m. to a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. {ZYm [av g_` : 3 KÊQ>o A{YH$V_ A H$ : 90 Time allowed : 3 hours Maximum Marks : 90 30/2 1 P.T.O.

2 gm_mý` {ZX}e : (i) g^r àíz A{Zdm` h & (ii) Bg àíz-nì _ 31 àíz h Omo Mma IÊS>m A, ~, g Am a X _ {d^m{ov h & (iii) IÊS> A _ EH$-EH$ A H$ dmbo 4 àíz h & IÊS> ~ _ 6 àíz h {OZ_ go àë`oh$ 2 A H$ H$m h & IÊS> g _ 10 àíz VrZ-VrZ A H$m Ho$ h & IÊS> X _ 11 àíz h {OZ_ go àë`oh$ 4 A H$ H$m h & (iv) H $bhw$boq>a H$m à`moj d{o V h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections A, (iii) (iv) B, C and D. Section A contains 4 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. Use of calculators is not permitted. àíz g»`m 1 go 4 VH$ àë`oh$ àíz 1 A H$ H$m h & IÊS> A SECTION A Question numbers 1 to 4 carry 1 mark each. 1. AmH ${V 1 _, O H $Ð dmbo d Îm H$s PQ EH$ Ordm h VWm PT EH$ ñne aoim h & `{X QPT = 60 h, Vmo PRQ kmv H$s{OE & AmH ${V 1 30/2 2

3 In Figure 1, PQ is a chord of a circle with centre O and PT is a tangent. If QPT = 60, find PRQ. Figure 1 2. `{X {ÛKmV g_rh$au px px + 15 = 0 Ho$ Xmo g_mz _yb hm, Vmo p H$m _mz kmv H$s{OE & If the quadratic equation px 2 2 then find the value of p. 5 px + 15 = 0 has two equal roots, 3. AmH ${V 2 _, EH$ _rzma AB H$s D±$MmB 20 _rq>a h Am a BgH$s ^y{_ na nan>mb BC H$s bå~mb 20 3 _rq>a h & gy` H$m CÞVm e kmv H$s{OE & AmH ${V 2 In Figure 2, a tower AB is 20 m high and BC, its shadow on the ground, is 20 3 m long. Find the Sun s altitude. Figure 2 30/2 3 P.T.O.

4 4. Xmo {^Þ nmgm H$mo EH $gmw CN>mbm J`m & XmoZm nmgm Ho$ D$nar Vbm na AmB g»`mam H$m JwUZ\$b 6 AmZo H$s àm{`h$vm kmv H$s{OE & Two different dice are tossed together. Find the probability that the product of the two numbers on the top of the dice is 6. IÊS> ~ SECTION B àíz g»`m 5 go 10 VH$ àë`oh$ àíz 2 A H H$m h & Question numbers 5 to 10 carry 2 marks each. 5. `{X {~ÝXþ A(x, y), B( 5, 7) VWm C( 4, 5) ñ maoir` hm, Vmo x VWm y _ gå~ýy kmv H$s{OE & Find the relation between x and y if the points A(x, y), B( 5, 7) and C( 4, 5) are collinear. 6. EH$ g_m Va lo T>r Ho$ àw_ n nxm Ho$ `moj\$b H$mo S n Ûmam Xem `m OmVm h & Bg lo T>r _ `{X S 5 + S 7 = 167 VWm S 10 = 235 h, Vmo g_m Va lo T>r kmv H$s{OE & In an AP, if S 5 + S 7 = 167 and S 10 = 235, then find the AP, where S n denotes the sum of its first n terms. 7. AmH ${V 3 _, Xmo ñne aoime± RQ VWm RP d Îm Ho$ ~mø {~ÝXþ R go ItMr JB h & d Îm H$m Ho$ÝÐ O h & `{X PRQ = 120 h, Vmo {gõ H$s{OE {H$ OR = PR + RQ. AmH ${V 3 30/2 4

5 In Figure 3, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If PRQ = 120, then prove that OR = PR + RQ. Figure 3 8. AmH ${V 4 _, 3 go_r {ÌÁ`m dmbo EH$ d Îm Ho$ n[ajv EH$ {Ì^wO ABC Bg àh$ma ItMm J`m h {H$ aoimiês> BD VWm DC H$s b ~mb`m± H«$_e 6 go_r VWm 9 go_r h & `{X ABC H$m joì\$b 54 dj go_r h, Vmo ^womam AB VWm AC H$s bå~mb`m± kmv H$s{OE & AmH ${V 4 In Figure 4, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. If the area of ABC is 54 cm 2, then find the lengths of sides AB and AC. Figure 4 30/2 5 P.T.O.

6 9. {ZåZ {ÛKmV g_rh$au H$mo x Ho$ {be hb H$s{OE : 4x 2 + 4bx (a 2 b 2 ) = 0 Solve the following quadratic equation for x : 4x 2 + 4bx (a 2 b 2 ) = `{X A(4, 3), B( 1, y) VWm C(3, 4) EH$ g_h$mou {Ì^wO ABC Ho$ erf h, {Og_ A na g_h$mou h, Vmo y H$m _mz kmv H$s{OE & If A(4, 3), B( 1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y. IÊS> g SECTION C àíz g»`m 11 go 20 VH$ àë`oh$ àíz 3 A H$ H$m h & Question numbers 11 to 20 carry 3 marks each. 11. AMmZH$ ~m T> AmZo na, Hw$N> H$ë`mUH$mar g ñwmam Zo {_b H$a gah$ma H$mo Cgr g_` 100 Q> Q> bjdmzo Ho$ {be H$hm VWm Bg na AmZo dmbo IM H$m 50% XoZo H$s noeh$e H$s & `{X àë`oh$ Q> Q> H$m {ZMbm ^mj ~obzmh$ma h {OgH$m ì`mg 4. 2 _r. h VWm D±$MmB 4 _r. h VWm D$nar ^mj Cgr ì`mg H$m e Hw$ h {OgH$s D±$MmB 2. 8 _r. h, Am a Bg na bjzo dmbo H $Zdg H$s bmjv < 100 à{v dj _r. h, Vmo kmv H$s{OE {H$ BZ g ñwmam H$mo {H$VZr am{e XoZr hmojr >& BZ g ñwmam Ûmam {H$Z _yë`m H$m àxe Z {H$`m J`m? [ = 7 22 br{oe ] Due to sudden floods, some welfare associations jointly requested the government to get 100 tents fixed immediately and offered to contribute 50% of the cost. If the lower part of each tent is of the form of a cylinder of diameter 4. 2 m and height 4 m with the conical upper part of same diameter but of height 2. 8 m, and the canvas to be used costs < 100 per sq. m, find the amount, the associations will have to pay. What values are shown by these associations? [Use = 7 22 ] 30/2 6

7 12. YamVb Ho$ EH$ {~ÝXþ A go EH$ hdmb OhmµO H$m CÞ`Z H$moU 60 h & 15 goh$ês H$s C S>mZ Ho$ nímmv², CÞ`Z H$moU 30 H$m hmo OmVm h & `{X hdmb OhmµO EH$ {ZpíMV D±$MmB _rq>a na C S> ahm hmo, Vmo hdmb OhmµO H$s J{V {H$bmo_rQ>a/K Q>m _ kmv H$s{OE & The angle of elevation of an aeroplane from a point A on the ground is 60. After a flight of 15 seconds, the angle of elevation changes to 30. If the aeroplane is flying at a constant height of m, find the speed of the plane in km/hr. 13. EH$ AÕ Jmobr` ~V Z H$m AmÝV[aH$ ì`mg 36 go_r h & `h Vab nxmw go ^am h & Bg Vab H$mo 72 ~obzmh$ma ~movbm _ S>mbm J`m h & `{X EH$ ~obzmh$ma ~movb H$m ì`mg 6 go_r hmo, Vmo àë`oh$ ~movb H$s D±$MmB kmv H$s{OE, O~{H$ Bg {H«$`m _ 10% Vab {Ja OmVm h & A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer. 14. EH$ Oma _ Ho$db bmb, Zrbr VWm Zma Jr a J H$s J X h & `mñàn>`m EH$ bmb a J H$s J X Ho$ {ZH$mbZo H$s àm{`h$vm 4 1 h & Bgr àh$ma Cgr Oma go `mñàn>`m EH$ Zrbr J X Ho$ {ZH$mbZo H$s àm{`h$vm 3 1 h & `{X Zma Jr a J H$s Hw$b J X 10 h, Vmo ~VmBE {H$ Oma _ Hw$b {H$VZr J X h & The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is 4 1. The probability of selecting a blue ball at random from the same jar is 3 1. If the jar contains 10 orange balls, find the total number of balls in the jar go_r ^wom dmbo EH$ KZmH$ma ãbm H$ Ho$ D$na EH$ AY Jmobm aim hþam h & AY Jmobo H$m A{YH$V_ ì`mg Š`m hmo gh$vm h? Bg àh$ma ~Zo R>mog Ho$ g nyu n ð>r` joì H$mo n Q> H$admZo H$m < 5 à{v 100 dj go_r H$s Xa go ì`` kmv H$s{OE & [ = br{oe ] A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of the solid so formed, at the rate of < 5 per 100 sq. cm. [ Use = ] 30/2 7 P.T.O.

8 16. `{X ( 2, 2) VWm (2, 4) H«$_e {~ÝXþ A VWm B Ho$ {ZX}em H$ h, Vmo {~ÝXþ P Ho$ {ZX}em H$ kmv H$s{OE O~{H$ P aoimiês> AB na h VWm AP = 7 3 AB. If the coordinates of points A and B are ( 2, 2) and (2, 4) respectively, find the coordinates of P such that AP = 7 3 AB, where P lies on the line segment AB go_r ì`mg VWm 3 go_r D±$Mo 504 e Hw$Am H$mo {nkbmh$a EH$ YmpËdH$ Jmobm ~Zm`m J`m & Jmobo H$m ì`mg kmv H$s{OE & AV BgH$m n ð>r` joì\$b kmv H$s{OE & [ = 7 22 br{oe ] 504 cones, each of diameter 3. 5 cm and height 3 cm, are melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area. [Use = 7 22 ] 18. EH$ g_mvw^w O Ho$ g^r erf EH$ d Îm na pñwv h & `{X Bg d Îm H$m joì\$b 1256 dj go_r h, Vmo g_mvw^w O H$m joì\$b kmv H$s{OE & [ = br{oe ] All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if the area of the circle is 1256 cm 2. [ Use = ] 19. x Ho$ {be hb H$s{OE : 2x x 60 = 0 Solve for x : 2x x 60 = EH$ g_mýva lo T>r H$m 16dm± nx BgHo$ Vrgao nx H$m nm±m JwZm h & `{X BgH$m 10dm± nx 41 h, Vmo BgHo$ àw_ 15 nxm H$m `moj\$b kmv H$s{OE & The 16 th term of an AP is five times its third term. If its 10 th term is 41, then find the sum of its first fifteen terms. 30/2 8

9 IÊS> X SECTION D àíz g»`m 21 go 31 VH$ àë`oh$ àíz 4 A H$ H$m h & Question numbers 21 to 31 carry 4 marks each. 21. {gõ H$s{OE {H$ d Îm H$s {H$gr Mmn Ho$ _Ü`-{~ÝXþ na ItMr JB ñne aoim, Mmn Ho$ A Ë` {~ÝXþþAm H$mo {_bmzo dmbr Ordm Ho$ g_m Va hmovr h & Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc. 22. EH$ Prb _ nmzr Ho$ Vb go 20 _rq>a D±$Mo {~ÝXþ A go, EH$ ~mxb H$m CÞ`Z H$moU 30 h & Prb _ ~mxb Ho$ à{v{~å~ H$m A go AdZ_Z H$moU 60 h & A go ~mxb H$s Xÿar kmv H$s{OE & At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30. The angle of depression of the reflection of the cloud in the lake, at A is 60. Find the distance of the cloud from A. 23. AÀN>r Vah go \ $Q>r JB EH$ Vme H$s JÈ>r go EH$ nîmm `mñàn>`m {ZH$mbm J`m & àm{`h$vm kmv H$s{OE {H$ {ZH$mbm J`m nîmm (i) (ii) (iii) (iv) hþhw$_ H$m nîmm h `m EH$ B $m h & EH$ H$mbo a J H$m ~mxemh h & Z Vmo Jwbm_ h VWm Z hr ~mxemh h & `m Vmo ~mxemh h `m ~oj_ h & A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is (i) (ii) (iii) (iv) a card of spade or an ace. a black king. neither a jack nor a king. either a king or a queen. 30/2 9 P.T.O.

10 24. AmH ${V 5 _, PQRS EH$ djm H$ma bm Z h {OgH$s ^wom PQ = 42 _rq>a h & Xmo d ÎmmH$ma \y$bm H$s Š`m[a`m± ^wom PS VWm QR na h {OZH$m Ho$ÝÐ Bg dj Ho$ {dh$um] H$m à{vàn>oxz {~ÝXþ O h & XmoZm \y$bm H$s Š`m[a`m (N>m`m {H$V ^mj) H$m Hw$b joì\$b kmv H$s{OE & AmH ${V 5 In Figure 5, PQRS is a square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flower beds (shaded parts). Figure EH$ R>mog YmVw Ho$ ~obz Ho$ XmoZmo {H$Zmam go Cgr ì`mg Ho$ AÕ Jmobo Ho$ ê$n _ YmVw {ZH$mbr JB & ~obz H$s D±$MmB 10 go_r VWm BgHo$ AmYma H$s {ÌÁ`m 4. 2 go_r h & eof ~obz H$mo {nkbmh$a 1. 4 go_r _moq>r ~obzmh$ma Vma ~ZmB JB & Vma H$s bå~mb kmv H$s{OE & [ = 7 22 br{oe ] From each end of a solid metal cylinder, metal was scooped out in hemispherical form of same diameter. The height of the cylinder is 10 cm and its base is of radius 4. 2 cm. The rest of the cylinder is melted and converted into a cylindrical wire of 1. 4 cm thickness. Find the length of 22 the wire. [Use = ] 7 30/2 10

11 26. EH$ Am`VmH$ma IoV H$m {dh$u BgH$s N>moQ>r ^wom go 16 _rq>a A{YH$ h & `{X BgH$s ~ S>r ^wom N>moQ>r ^wom go 14 _rq>a A{YH$ h, Vmo IoV H$s ^womam H$s bå~mb`m± kmv H$s{OE & The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field. 27. g_m Va lo T>r 8, 10, 12,... H$m 60dm± nx kmv H$s{OE, `{X Cg_ Hw$b 60 nx h & AV Bg lo T>r Ho$ A {V_ 10 nxm H$m `moj\$b kmv H$s{OE & Find the 60 th term of the AP 8, 10, 12,..., if it has a total of 60 terms and hence find the sum of its last 10 terms. 28. EH$ ~g nhbo 75 {H$bmo_rQ>a H$s Xÿar {H$gr Am gv Mmb go MbVr h VWm CgHo$ ~mx H$s 90 {H$bmo_rQ>a H$s Xÿar nhbo go 10 {H$bmo_rQ>a à{v K Q>m A{YH$ H$s Am gv Mmb go MbVr h & `{X Hw$b Xÿar 3 K Q>o _ nyar hmovr h, Vmo ~g H$s nhbr Mmb kmv H$s{OE & A bus travels at a certain average speed for a distance of 75 km and then travels a distance of 90 km at an average speed of 10 km/h more than the first speed. If it takes 3 hours to complete the total journey, find its first speed. 29. {gõ H$s{OE {H$ d Îm Ho$ {H$gr {~ÝXþ na ñne aoim ñne {~ÝXþ go OmZo dmbr {ÌÁ`m na b ~ hmovr h & Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 30. EH$ g_h$mou {Ì^wO ABC H$s amzm H$s{OE, {Og_ AB = 6 go_r, BC = 8 go_r VWm B = 90 h & B go AC na b ~ BD It{ME & {~ÝXþAm B, C VWm D go hmoh$a OmZo dmbm EH$ d Îm It{ME VWm A go Bg d Îm na ñne aoimam H$s amzm H$s{OE & Construct a right triangle ABC with AB = 6 cm, BC = 8 cm and B = 90. Draw BD, the perpendicular from B on AC. Draw the circle through B, C and D and construct the tangents from A to this circle. 31. k Ho$ _mz kmv H$s{OE {OZgo (k+1, 1), (4, 3) VWm (7, k) erfm] dmbo {Ì^wO H$m joì\$b 6 dj BH$mB hmo &$ Find the values of k so that the area of the triangle with vertices (k+1, 1), (4, 3) and (7, k) is 6 sq. units. 30/2 11 P.T.O.

II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS

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